We investigate the classical Gilmore-Lawler lower bound for the quadratic assignment problem. We provide evidence of the difficulty of improving the Gilmore-Lawler Bound and develop new bounds by means of optimal reduction schemes. Computational results are reported indicating that the new lower bounds have advantages over previous bounds and can be used in a branch-and-bound type algorithm for...

“The Baron’s omni-sequence”, B(n), first defined by Khovanova and Lewis (2011), is a sequence that gives for each n the minimum number of weighings on balance scales that can verify the correct labeling of n identically-looking coins with distinct integer weights between 1 gram and n grams. A trivial lower bound on B(n) is log3 n, and it has been shown that B(n) is log3 n + O(log log n). In thi...

We consider a bilevel optimization approach for parameter learning in nonsmooth variational models. Existing approaches solve this problem by applying implicit differentiation to a sufficiently smooth approximation of the nondifferentiable lower level problem. We propose an alternative method based on differentiating the iterations of a nonlinear primal–dual algorithm. Our method computes exact...

One of the most important quantum algorithms ever discovered is Grover’s algorithm for searching an unordered set. We give a new lower bound in the query model which proves that Grover’s algorithm is exactly optimal. Similar to existing methods for proving lower bounds, we bound the amount of information we can gain from a single oracle query, but we bound this information in terms of angles. T...

In this paper an efficient Algorithm based on Zadeh's extension principle has been investigated to approximate fuzzy solution of two-point fuzzy boundary value problems, with fuzzy boundary values. We use finite difference method in term of the upper bound and lower bound of $r$- level of fuzzy boundary values. The proposed approach gives a linear system with crisp tridiagonal coefficients matr...

In the paper, the Fenchel problem of level sets in the smooth case is solved by deducing a new and “nice” geometric necessary and sufficient condition for the existence of a smooth convex function with the level sets of a given smooth pseudoconvex function. The main theorem is based on a general differential geometric tool, the space of paths defined on smooth manifolds. This approach provides ...

Let Gn be the subgraph of the hypercube Qn induced by levels between k and n − k, where n ≥ 2k + 1 is odd. The well-known middle level conjecture asserts that G2k+1 is Hamiltonian for all k ≥ 1. We study this problem in Gn for fixed k. It is known that Gn and G 1 n are Hamiltonian for all odd n ≥ 3. In this paper we prove that also Gn is Hamiltonian for all odd n ≥ 5, and we conjecture that Gn ...

In this paper, we present a mathematical framework for the problem of minimizing the cash-out penalties of a natural gas shipper. The problem is modeled as a mixedinteger bilevel programming problem having one Boolean variable in the lower level problem. Such problems are difficult to solve. To obtain a more tractable problem we move the Boolean variable from the lower to the upper level proble...